In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines is the same for all but one exceptional parallel class of lines. We call such sets regular of affine type. When the lines of the exceptional parallel class have the same intersection numbers, then we call these sets regular of pointed type. Classical examples are e.g. unitals; a detailed study and constructions of such sets with few intersection numbers is due to Hirschfeld and Sz\H{o}nyi from 1991. We here provide some general construction methods for regular sets and describe a few infinite families. The members of one of these families have the size of a unital and meet affine lines of $\mathrm{PG}(2, q^2)$ in one of $4$ possible intersection numbers, each of them congruent to $1$ modulo $\sqrt{q}$. As a byproduct, we determine the intersection sizes of the Hermitian curve defined over $\mathrm{GF}(q^2)$ with suitable rational curves of degree $\sqrt{q}$ and we obtain $\sqrt{q}$-divisible codes with $5$ non-zero weights. We also determine the weight enumerator of the codes arising from the general constructions modulus some $q$-powers.

翻译：本文研究了有限Desarguesian平面中的点集，这些点集的直线交点数多重集在除一条例外平行直线类外均相同。我们将此类点集称为仿射型正则集。当例外平行直线类中的直线具有相同交点数时，则称为点型正则集。经典例子包括单位线（unital）；Hirschfeld与Sz\H{o}nyi于1991年对具有少交点数的此类集合进行了详细研究并构造。本文提出了一些正则集的一般构造方法，并描述了若干无限族。其中一族成员的尺寸与单位线相同，且与$\mathrm{PG}(2, q^2)$中仿射线的交点仅取$4$种可能值，每种值均与$1$模$\sqrt{q}$同余。作为副产品，我们确定了定义在$\mathrm{GF}(q^2)$上的Hermitian曲线与次数为$\sqrt{q}$的合适有理曲线的交点大小，并得到了具有$5$个非零权重的$\sqrt{q}$-可除码。我们还确定了由一般构造模某些$q$幂次所得到码的权重枚举器。